Mathematics Basic flashcards

35 cards. Read the term and the definition here. The deck at the top of this page is optional.

Absolute value

The distance of a number from zero on the number line. It is never negative: |−7| = 7 and |7| = 7, and |x| = 0 only when x = 0.

Scientific notation

A number written as a × 10^n where 1 ≤ |a| < 10 and n is an integer. For example, 45,000 is written 4.5 × 10^4.

Rational number

A number expressible as a fraction of two integers with nonzero denominator. Its decimal form either terminates or repeats, unlike irrational numbers such as √2 or π.

Negative exponent rule

For any nonzero base a and positive integer n, a^(−n) equals 1/a^n. Thus 2^(−3) = 1/8, and a negative exponent indicates a reciprocal, not a negative result.

Percent change

Computed as (new value − old value) divided by old value, times 100. A positive result is an increase; a negative result is a decrease.

Unit conversion by cancellation

Multiply by conversion factors written as fractions equal to one, arranging units so unwanted units cancel. Converting 3 hours uses 3 h × (60 min / 1 h) = 180 min.

Slope formula

For two points (x₁, y₁) and (x₂, y₂), slope m = (y₂ − y₁)/(x₂ − x₁). It measures vertical change per unit of horizontal change; vertical lines have undefined slope.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic, including ones that do not factor.

Discriminant

The expression b² − 4ac from the quadratic formula. Positive means two real solutions, zero means exactly one, and negative means no real solutions.

Inequality reversal rule

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. For example, −2x > 6 becomes x < −3.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so their product is −1. A line with slope 2 is perpendicular to one with slope −1/2.

Point-slope form

A linear equation written y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line. It builds an equation from a slope and one point.

System with no solution

A system of two linear equations has no solution when the lines are parallel with different intercepts, meaning the slopes are equal and the constants differ.

Logarithm product rule

For positive arguments, log(xy) = log x + log y. The logarithm of a product equals the sum of the logarithms, which converts multiplication into addition.

Function definition

A rule assigning exactly one output to each input in its domain. A graph represents a function when no vertical line crosses it more than once.

Domain restriction

Values excluded from a function's inputs. Denominators cannot equal zero, and even roots require non-negative radicands, so f(x) = √x requires x ≥ 0.

Exponential growth versus decay

In f(x) = a·b^x, the function grows when b > 1 and decays when 0 < b < 1. The value a is the starting amount when x = 0.

Horizontal graph shift

Replacing x with (x − h) shifts a graph h units right, while (x + h) shifts it h units left. Horizontal shifts act opposite to the sign inside the parentheses.

Inverse function

A function that reverses another, written f⁻¹. It satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, and its graph reflects the original across the line y = x.

Even function

A function satisfying f(−x) = f(x) for every input in its domain. Its graph is symmetric about the y-axis, as with f(x) = x².

Average rate of change

For f on the interval from a to b, it equals (f(b) − f(a)) / (b − a). Geometrically it is the slope of the secant line through the two points.

Composition of functions

The function (f ∘ g)(x) = f(g(x)), applying g first and then f. Composition is generally not commutative, so f(g(x)) may differ from g(f(x)).

Pythagorean theorem

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c². Its converse identifies right triangles.

Circle area and circumference

A circle of radius r has area A = πr² and circumference C = 2πr. Diameter d = 2r, so the circumference also equals πd.

Interior angle sum

The interior angles of a polygon with n sides sum to (n − 2) × 180°. A quadrilateral therefore totals 360° and a pentagon totals 540°.

Similar figure area ratio

When two figures are similar with scale factor k, corresponding lengths scale by k and areas scale by k². Doubling all side lengths multiplies area by four.

Cone volume

The volume of a cone is V = ⅓πr²h, one third of the cylinder with the same base and height. A sphere has volume (4/3)πr³.

Distance formula

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It follows directly from the Pythagorean theorem on the coordinate plane.

Angle pair relationships

Complementary angles sum to 90°, supplementary angles sum to 180°, and vertical angles formed by intersecting lines are equal. Angles on a straight line total 180°.

Mean, median, mode

The mean is the sum divided by the count, the median is the middle ordered value, and the mode is the most frequent value. Outliers affect the mean most.

Standard deviation

A measure of how far values typically lie from the mean. A larger standard deviation means greater spread; adding the same constant to every value leaves it unchanged.

Complement rule

For any event A, the probability of A not occurring is P(not A) = 1 − P(A). The two probabilities always sum to exactly one.

Independent events

Events are independent when one occurring does not change the other's probability. Then P(A and B) = P(A) × P(B).

Conditional probability

The probability of A given that B occurred: P(A | B) = P(A and B) / P(B), provided P(B) > 0. It restricts the sample space to event B.

Expected value

The long-run average of a random variable, found by multiplying each outcome by its probability and summing the products. It need not equal any actual outcome.